From the given options, the city that has been the venue of the Asian Games for the maximum number of times from the year 1951 to the year 2006 is Bangkok, which is Option 2.

Here is the detailed reasoning:

1. Delhi: Delhi, the capital city of India, hosted the Asian Games in 1951 and again in 1982. It has hosted the event twice in total.

2. Bangkok: Bangkok, the capital city of Thailand, has been a prominent host for the Asian Games. It has hosted the event a total of four times. Bangkok first hosted the Asian Games in 1966, then again in 1970, 1978, and 1998. The city`s infrastructure, sporting facilities, and experience in hosting major international events have contributed to its repeated selection as the venue.

3. Tokyo: Tokyo, the capital city of Japan, has hosted the Asian Games once, in 1958. While Tokyo has been a host for various international sporting events, it has not been the venue for the Asian Games on multiple occasions.

4. Beijing: Beijing, the capital city of China, hosted the Asian Games once in 1990. Beijing has since hosted other major international sporting events, such as the 2008 Summer Olympics, but it has not been selected as the Asian Games venue for multiple editions.

Based on the information provided, Bangkok has hosted the Asian Games the most times (four times) from 1951 to 2006. Therefore, Option 2: Bangkok is the correct answer.